Langmuir 1990,6, 691-694 layer. Furthermore, the nitrile surface is not hydrophilic, but is permeable. These results suggest the importance of steric and electrostatic interactions between appended functional groups in determining the packing of the chains. Recognizing the sensitivity of barrier properties to such effects, we now intend to take advantage of the synthetic scope of organic chemistry to control the placement of electroactive species at the electrochemical interface and simultaneously to control the packing and hence barrier properties of the monolayer.
Acknowledgment. We thank Ralph Nuzzo for the use of his equipment, extensive technical advice, and valu-
691
able scientific discussions; Dennis Trevor for the use of his STM; and Tony Mujsce for acquiring GC-MS data. C.C. thanks Dave Allara for introducing him to reflection spectroscopy and Ian Robinson for discussions of electron diffraction.
Supplementary Material Available: Text giving details of the syntheses and analyses of the thiols and the procedures for the preparation and analysis of the gold substrates and the
monolayer films; Table S1, vibrational assignments, frequencies, and absorption intensities; Figure S1,an STM image of Au on Ti on Si; and Figure S2,infrared absorption spectra from 1800 to 3800 cm-' (14 pages). Ordering information is given on any current masthead page.
General Equations for Describing Diffusion on the Heterogeneous Surface at Finite Coverages V. Pereyra and G. Zgrablich Instituto de Investigaciones en Tecnologia Quimica, Universidad Nacional de Sun Luis, CONICET, Casilla de Correo 290, 5700 Sun Luis, Argentina
V. P. Zhdanov* Institute of Catalysis, Novosibirsk 630090, U S S R Received May 1, 1989. In Final Form: August 16, 1989 General equations are derived for describing diffusion on the heterogeneous surface in the framework of the lattice-gas model. The effect of the surface heterogeneity and lateral interactions between adsorbed particles on the coverage dependence of the chemical diffusion coefficient is discussed.
Introduction Surface diffusion is of considerable intrinsic interest and is also important for understanding the mechanism of surface reactions, ranging from the simplest, such as recombination of adsorbed particles, to the complex processes encountered in heterogeneous Its intrinsic interest arises from the dynamical and statistical features of particles in adsorbed overlayers. For this reason, surface diffusion has received the attention of many researchers in the last decade. In particular, the effect of lateral interactions between adsorbed particles on diffusion over the uniform surface at finite coverages is considered in detail.2 Diffusion on the heterogeneous surface at low coverages (the random-walk models) is also well ~ t u d i e d .Our ~ objective is to investigate diffusion on the heterogeneous surface at finite coverages. In this case, diffusion is simultaneously affected by the surface heterogeneity and by lateral interactions. Our attention is centered on periodic lattices with random transition rates. Transport processes in topologically disordered systems are not considered. (1)Ehrlich, G.; Stolt, K. Ann. Reo. Phys. Chem. 1980,31,603. Gomer, R. Vacuum 1983, 33, 537. Naumovets, A. G.; Vedula, Yu. S. Surface Sci. Rep. 1985,4, 365. Doll, J. D.; Voter, A. F. Ann. Reu. Phys. Chem. 1987,38,413. (2) Zhadanov, V. P.; Zamaraev, K. I. Usp. Fir.Nauk 1986, 149, 635 (English translation: Sou. Phys. Usp.1986, 29, 755). (3) Haus, J. W.; Kehr, K. W. Phys. Rep. 1987,150, 264.
0743-7463/90/2406-0691$02.50/0
A similar problem has been recently studied by using Monte Carlo simulations of self-diffusion4 and collective diffusion5 of particles without lateral interactions. In our paper, we discuss the coverage dependence of the chemical diffusion coefficient taking into account lateral interactions. Besides, in comparison with ref 4 and 5 , we use a more general expression connecting a saddle point energy with energies of nearest-neighbor sites (see eq 7). Our main assumptions are as follows. Adsorbed particles A are located in a two-dimensional array of surface sites. For clarity, we consider a square lattice. A single type of sites is used. A given site is either vacant or occupied by a single adsorbed particle. The surface heterogeneity is caused by the energy distribution of sites. Diffusion occurs via activated jumps of particles to nearestneighbor empty sites. Activated particles A* are located on the boundary between two nearest-neighbor sites. Two neighbor sites, occupied by an activated particle (i.e., by an activated complex), cannot be occupied by other particles. Lateral interactions between particles are taken into consideration. It is also assumed that activated complexes interact with neighboring adsorbed particles. Repulsive interactions are assigned positive values. More(4) Kirchheim, R.; Stolz, U. Acta Metall. 1987, 35, 281 and references therein. (5) Mak, C. H.; Anderson, H. C.; George, S. M. J. Chem. Phys. 1988,
88, 4052.
0 1990 American Chemical Society
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Pereyra et al.
over, temperature is assumed to be higher than the critical one of the phase transitions in the adsorbed overlayer. Our approach to the problem is based upon application of the grand canonical distribution. This approach has been used recently to describe diffusion on the uniform surface without6 and with7 steps. Similar ideas and the percolation theory have been also used* to study diffusion on the heterogeneous surface. However, the results* are rather limited. Our results are more general.
I
a
General Equations The jump rate of a particle is dependent on energy of sites and on configuration of surrounding particles. In general, the flux of particles from row 1 to row 2 can be represented as J1,Z
= Nl( x P A O , i k A O , i )
(1)
i
where N , = l / a is the number of sites on a unity measure of length, a is the lattice spacing, PAo,iis the probability that the site in the first row is occupied by a particle A and the nearest-neighbor site in the second row is empty, with the environment being marked by the index i, and kAOiis the rate constant for the transition AO,i OA,i. The brackets mean an average over site energies. Using the grand canonical distribution, we have
-
(2) PA0.i = p W , i exp[(pl - tAO,i - El)/q where PDoo,i is the probability that a pair of the nearestneighbor sites is empty, with the environment marked by the index i, p1 is the chemical potential in row 1,E , is the energy of a site in the first row, tAO,jis the lateral interaction of a particle A with surrounding particles, and the Boltzmann constant is set to unity. Substitution of eq 2 into eq 1 yields I
(3)
By analogy, the flux from row 2 to row 1 is 1
(4) where p2 is the chemical potential in row 2. According to a detailed balancing principle, we have kAO,i
exp[-(tAO,i +
= kOA,i exP[-(tOA,i + &)/TI
(5)
Hence, the sum in eq 3 is the same as that in eq 4. Thus, the total flux, J = J1,,- J2,1,is expressed through a gradient in the chemical potential, and we derive the following expression for the chemical diffusion coefficient:
where 0 is a surface coverage. So far, our analysis is applicable both to activated and tunnel diffusion. For activated diffusion, it seems reasonable to write
(6)Zhdanov, V. P. Surf. Sci. 1985, 149,L13;1986, 177,L896; 1988, 194,1; Phys. Lett. 1989, 137,225. (7)Zhdanov, V . P. Phys. Lett. 1989, 137,409. (8) Pereyra, V.;Zgrablich, G. Surf. Sci. 1989, 209, 512.
DIFFUSION COORDINATE
Figure 1. Schematic topology of the potential energy for diffusion at different values of a.
where u is a preexponential factor, E,(O) is the activation energy for the transition between two nearest-neighbor sites with E, = E, = 0 provided that other nearest sites are empty, ti* - cAO and aE, + aE2- E, are the variation of the activation energy due to lateral interactions and the surface heterogeneity, respectively (aE, + aE, is the variation of the top of the potential barrier), ti* is the lateral interaction of the activated complex with its environment, and a > 0 is a parameter. Different values of the parameter a correspond to different topologies of the potential energy for diffusion (Figure 1). If a = 0 (Figure la), the potential energy for diffusion is like that of a “random traps lattice” (RTL), a lattice of traps with randomly distributed depths; while a t a = 1/2 (Figure IC),the potential for diffusion is a ”continuum random perturbation” (CRP) of a periodic potential. The case a = 1 / 3 (Figure l b ) is intermediate between these two important representations. Substitution of eq 7 into eq 6 yields
D=
exp(p/T)-1 -S aP
T a0
with
where ueff = u exp(-E,(O)/T). Accurate calculation of the coverage dependence of the chemical potential and the sum S is a difficult problem. Simple expressions may be derived only in the meanfield approximation. This approximation yields
s = exp[-(Zz - Z ) ~ , * O / T ~ S , ~
(10)
S, = I f ( E ) exp(-aE/T)[l- 0,(E,0)]dE
(11)
where 22 - 2 is the number of neighboring sites for the activated complex, z = 4 is the number of nearestneighbor sites for one site, el* is the A*-A lateral inter-
Langmuir, Vol. 6, No. 3, 1990 693
Diffusion on the Heterogeneous Surface action, f(E) is the energy distribution of sites, and B,(E,O)is the probability that a site with energy E is filled Os = em[(* - E - ztlO)/T‘l/{l
+ exp[(lr.- E - zt16)/TlI
(12) is the A-A lateral interaction. The surface coverage 8 and the probability O,(E,8) are connected as t1
8 = Sf(E)O,(E,O)dE
(13)
Using eq 12 and 13, we can calculate the coverage dependence of the chemical potential and then calculate sums S,, S, and the chemical diffusion coefficient.
Diffusion at Low and High Coverages It is of interest to analyze diffusion a t low and high coverages. In the former case (at O 0 result in an increase in potential barriers for diffusion and lead to a decrease in the diffusion coefficient with increasing coverage. Attractive interactions el* C 0 result in an increase in the diffusion coefficient with increasing coverage. The coverage dependence defined by eq 17 is very simple. For this reason, this dependence is neglected in our following calculations (i.e., we assume that el* = 0). To calculate the diffusion coefficient, we have employed
c
c.?
c.4
c 6
0.5
a
c
0.2
3.1
C.6
2.6
e
Figure 2. Diffusion coefficient as a function of coverage for the rectangular energy distribution of sites.
the rectangular energy distribution of sites
f ( E ) = 1/2Aat-A< E < A =OatE A In this case
(18)
exp(ztl8/T)[exp(2OA/n - 11 (19) exp(A/T) - exp[(28 - UA/V The results of calculations are shown in Figure 2. We have calculated the diffusion coefficient also for the Gaussian energy distribution of sites exp(P/T) =
f(E) = e ~ p ( - E ~ / 2 ( r ~ ) / [ ( 2 x ) ” ~ 0 . ] (20) In order to avoid unphysical situations, we have taken minimum and maximum values for energies of sites and renormalized the Gaussian distribution so that
EZf(E)dE = 1 As a rule, we have used Emin= -5T and E, = 5T. The coverage dependence of the diffusion coefficient for the Gaussian distribution (Figure 3) is almost the same as in the case of the rectangular distribution (Figure 2). Finally, we have calculated the diffusion coefficient for the log-normal energy distribution of sites
f ( E ) = exp[-(ln E -In E,)2/272]/[7E(2a)1/2] (21) where E, and T are the median and dispersion, respectively. Figure 4 shows the results for noninteracting particles. The coverage dependence of the chemical diffusion coefficient is seen (Figures 2a, 3a, and 4) to be very strong for the RTL model (CY= 0). This is explained as follows. A t low coverages, the low-energy sites are preferentially occupied. Particles, adsorbed on such sites, have a very low mobility due to high activation energy that they need to jump. As coverage increases, particles from sites with higher and higher energy (and hence with lower activation energy) make a contribution to migration, and the diffusion coefficient increases. This effect is somewhat smeared near the saturation coverage (e = 1). As soon as the surface differs from the RTL model, for example, for CY = 113 (Figures 2b and 3b), the energy of the “destination” sites, E,, begins t o influence the jump rate. This effect leads to an decrease in the coverage dependence of the diffusion coefficient. It is of interest that for the rectangular and Gaussian energy distributions a t a = 1/ 2 a maximum of the diffusion coefficient of noninteracting particles occurs a t 8 E 1/2, and the coverage depen-
Pereyra et al.
694 Langmuir, Vol. 6, No. 3, 1990 10
i
E , = c
0.2
~. ..,
c.5
t.5
1 8
c.4
C.6
0.6
e
1
Figure 4. Diffusion coefficient as a function of coverage for the log-normal energy distribution of sites with E J T = 4 and T I T = 0.7.
ficient. This is explained by a decrease (increase) in the potential barriers for diffusion jumps due to a rise (fall) I I I of the bottom of the potential energy of adsorbed particles. This effect is the same as in the case of the uniform 1attice.*g5 On the whole, the coverage dependence of the diffusion coefficient depends on a relative role of heterogeneity and lateral interactions. If the heterogeneity is rather considerable and lateral interaction between adsorbed particles are repulsive, the coverage dependence of the diffusion coefficient is very strong because both factors lead to an increase of the diffusivity with increasing coverage (see, e.g., Figure 2a a t zt,/T = 3 or Figure 3a at c / T = 4 , I! and z t J T = 2). If the heterogeneity is rather strong but I I I lateral interactions between adsorbed particles are attrac3 0.5 0.5 1 0.5 1 e tive, the coverage dependence of the diffusion coefficient can be weak because the different factors compenC d = ./2 sate each other (see, e.g., Figure 2 at zt,/T = -3 and Figd/r 1 d / 7 = 2 d/I'=s i ure 3a and 3b at a/T = 4 and ze,/T = -2). Finally, if i i the heterogeneity is weak and lateral adsorbate-adsorbate interactions are attractive, the diffusion coefficient decreases with increasing coverage (see, e.g., Figure 3 a t u / T = 1 and zt,/T = -2) due to a fall of the bottom of the potential energy of adsorbate particles. In summary, we have derived general equations for describing diffusion on the heterogeneous surface in the framework of the lattice-gas model. The effect of heterogeneity and lateral interactions on the coverage depen3 1 dence of the chemical diffusion coefficient has been dem7 ".5 1 c 5 onstrated and discussed. The coverage dependence of Figure 3. Diffusion coefficient as a function of coverage for the diffusion coefficient is shown to be strong or weak the Gaussian distribution of sites. depending on a relative role of heterogeneity and lateral interactions. The derived results can be used, for examdence of the diffusion coefficient is symmetric about this ple, to interpret experimental data that can be obtained value of coverage. This is a consequence of symmetry of for diffusion after burning a hole out of an adsorbed layer these distribution with respect to the average energy. In (at present, unfortunately, the data are obtained only for the case of the log-normal distribution (Figure 4),the uniform surfacesg). The outlined results can be used also presence of a maximum of the diffusion coefficient also to analyze the kinetics of reactions limited by surface differentiates the cases in which a > 0 from that of the diffusion (by analogy with ref 10). RTL model ( a = 0). However, now the symmetry about 6 = 112 is lost for a = 112. (9) Seebauer, E. G.; Kong, A. C. F.; Schmidt, L. D. J. Chem. Phys. Repulsive (attractive) adsorbate-adsorbate interac1988,88, 6597. (IO) Zhdanov, V. P. Surf. Sci. 1988, 195, L217. tions result in an increase (decrease) in the diffusion coef-
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